A necessary condition in a De Giorgi type conjecture for elliptic systems in infinite strips

Abstract

Given a bounded Lipschitz domain $\omega \subset {\mathbb{R}}^{d-1}$ and a lower semicontinuous function $W:{\mathbb{R}}^N\to {\mathbb{R}}_{+}\cup \{+\infty \}$ that vanishes on a finite set and that is bounded from below by a positive constant at infinity, we show that every map $u:\mathbb{R}\times \omega \to {\mathbb{R}}^N$ with \[ \int_{\mathbb{R}\timesω}\big(\lvert\nabla u\rvert^2+W(u)\big)\mathop{}\mathopen{}\mathrm{d} x_1\mathop{}\mathopen{}\mathrm{d}x'<+\infty\] has a limit $u^{\pm }\in \{W=0\}$ as $x_1\to \pm \infty$. The convergence holds in $L^2(\omega )$ and almost everywhere in $\omega$. We also prove a similar result for more general potentials $W$ in the case where the considered maps $u$ are divergence-free in $\mathbb{R}\times \omega$ with $\omega$ being the $(d-1)$-torus and $N=d$.